MATH 125 Calculus II
Contents
Integrals
Indefinite integrals
| Description | Equations |
|---|---|
| Indefinite integral (antiderivative) | $F(x) = \displaystyle\int f(x) \ dx \newline F'(x) = f(x)$ |
| Antiderivative as a family of functions (Plus $C$!) |
If $F$ is an antiderivative of $f$, $C$ is a constant, then the most general antiderivative is $F(x) + C$ |
Table of indefinite integrals
| Function $f(x)$ | Antiderivative $F(x)$ | Function $f(x)$ | Antiderivative $F(x)$ |
|---|---|---|---|
| $x^n$ | $\dfrac{x^{n+1}}{n+1}+C$ | $\dfrac{1}{x}$ | $\ln\lvert x \rvert + C$ |
| $e^x$ | $e^x + C$ | $b^x$ | $\dfrac{b^x}{\ln b} + C$ |
| $\sin x$ | $-\cos x + C$ | $\cos x$ | $\sin x + C$ |
| $\sec^2 x$ | $\tan x + C$ | $csc^2 x$ | $-\cot x + C$ |
| $\sec x\tan x$ | $\sec x + C$ | $\csc x\cot x$ | $-\csc x + C$ |
| $\dfrac{1}{x^2 + a^2}$ | $\dfrac{1}{a}\arctan \left(\dfrac{x}{a}\right) + C$ | $\dfrac{1}{\sqrt{a^2-x^2}}$ | $\arcsin \left(\dfrac{x}{a}\right) + C$ |
Definite integrals as Riemann sums
| Description | Equations |
|---|---|
| Area | $A = \lim\limits_{n\to\infin} R_{n} = \lim\limits_{n\to\infin} \sum\limits_{i=1}^{n}f(x_i)\Delta x$ |
| Definite integral | $\int_{a}^{b} f(x) \ dx = \lim\limits_{n\to\infin} \sum\limits_{i=1}^{n}f(x_i^*)\Delta x$ |
| Operational definition of definite integral as Riemann sum | $\int_a^b f(x) \ dx = \lim\limits_{n\to\infin} \sum\limits_{i=1}^n f(x_i)\Delta x$ $\Delta x = \frac{b-a}{n} \newline x_i = a+i\Delta x$ |
| Sums of powers of positive integers | $\sum\limits_{i=1}^{n}i = \frac{n(n+1)}{2} \newline \sum\limits_{i=1}^{n}i^2 = \frac{n(n+1)(2n+1)}{6} \newline \sum\limits_{i=1}^{n}i^3 = \left[ \frac{n(n+1)}{2} \right]^2$ |
| Properties of summation | $\sum\limits_{i=1}^{n}c = nc \newline \sum\limits_{i=1}^{n}ca_i = c\sum\limits_{i=1}^{n}a_i \newline \sum\limits_{i=1}^{n}(a_i \pm b_i) = \sum\limits_{i=1}^{n}a_i \pm \sum\limits_{i=1}^{n}b_i$ |
Properties of definite integrals
| Description | Equations |
|---|---|
| Reversing the bounds changes the sign of definite integrals | $\int_a^b f(x) \ dx = -\int_b^a f(x) \ dx$ |
| Definite integral is zero if upper and lower bounds are the same | $\int_a^a f(x) \ dx = 0$ |
| Definite integrals of constant | $\int_a^b c \ dx = c(b-a)$ |
| Addition and subtraction of definite integrals | $\int_a^b [f(x) \pm g(x)] \ dx \newline = \int_a^b f(x) \ dx \pm \int_a^b g(x) \ dx$ |
| Constant multiple of definite integrals | $\int_a^b cf(x) \ dx = c\int_a^b f(x) \ dx$ |
| Comparison properties of definite integrals | If $f(x) \ge 0$ for $x\in[a,b]$, then $\int_a^b f(x) \ dx \ge 0$ |
| Comparison properties of definite integrals | If $f(x) \ge g(x)$ for $x\in[a,b]$, then $\int_a^b f(x) \ dx \ge \int_a^b g(x) \ dx$ |
| Comparison properties of definite integrals | If $m \le f(x) \le M$ for $x\in[a,b]$, then $m(b-a) \le \int_a^b f(x) \ dx \le M(b-a)$ |
Fundamental theorem of calculus
| Description | Equations |
|---|---|
| Fundamental theorem of calculus I ($f$ is continuous on $[a,b]$) |
$g(x) = \displaystyle\int_a^x f(t) \ dt \newline g'(x) = f(x)$ |
| Fundamental theorem of calculus II ($f$ is continuous on $[a,b]$) |
$\displaystyle\int_a^b f(x) \ dx = F(b) - F(a)$ where $F$ is any antiderivative of $f$ |
| Net change theorem The integral of a rate of change is the net change |
$\displaystyle\int_a^b F'(x) \ dx = F(b) - F(a)$ |
Substitution rule
| Description | Equations |
|---|---|
| Substitution rule (u-substitution) $u \equiv g(x)$ |
$\displaystyle\int f(g(x)) g'(x) \ dx = \int f(u) \ du$ |
| Substitution rule for definite integrals $u \equiv g(x)$ |
$\displaystyle\int_a^b f(g(x))g'(x) \ dx = \int_{g(a)}^{g(b)} f(u) \ du$ |
| Integrals of even functions | $\int_{-a}^a f(x) \ dx = 2 \int_{0}^a f(x) \ dx$ |
| Integrals of odd functions | $\int_{-a}^a f(x) \ dx = 0$ |
Techniques of Integration
Integration by parts
| Description | Equations |
|---|---|
| Integration by parts | $\int f(x)g'(x) \ dx \newline = f(x)g(x) - \int g(x)f'(x) \ dx$ |
| Integration by parts | $\int u \ dv = uv - \int v \ du$ |
| Integration by parts for definite integrals | $\int_a^b fg' \ dx = [fg]_a^b - \int_a^b f’g \ dx$ |
Approximating integrals
| Description | Equations |
|---|---|
| Midpoint rule | $\int_a^b f(x) \ dx \approx \sum\limits_{i=1}^n f(\bar{x}_i)\Delta x$ $\Delta x = \frac{b-a}{n} \newline \bar{x}_i = \frac{1}{2}(x_{i-1}+x_i)$ |
| Error bound for midpoint rule | $\lvert E_M \rvert \le \dfrac{K(b-a)^3}{24n^2}$ |
| Trapezoidal rule | $\int_a^b f(x) \ dx \approx \frac{1}{2}\Delta x [f(x_0) + 2f(x_1) + … + 2f(x_{n-1}) + f(x_n)]$ $\Delta x = \frac{b-a}{n} \newline x_i = a + i\Delta x$ |
| Error bound for trapezoidal rule | $\lvert E_T \rvert \le \dfrac{K(b-a)^3}{12n^2}$ |
| Simpson’s rule | $\int_a^b f(x) \ dx \approx \frac{1}{3}\Delta x [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + … + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n)]$ $\Delta x = \frac{b-a}{n}$, n is even |
| Error bound for Simpson’s rule | $\lvert E_S \rvert \le \dfrac{K(b-a)^5}{180n^4}$ |
Trigonometric integrals
| Description | Equations |
|---|---|
| Integral of odd power of cosine $(u = \sin x)$ |
$\int \sin^m(x)\cos^{2k+1}(x) \ dx \newline = \int \sin^m(x) [\cos^2 (x)]^k \ dx \newline = \int \sin^m(x)[1-\sin^2(x)]^k \ dx$ |
| Integral of odd power of sine $(u = \cos x)$ |
$\int \sin^{2k+1}(x)\cos^{n}(x) \ dx \newline = \int [\sin^2 (x)]^k \cos^n(x) \sin(x) \ dx \newline = \int [1-\cos^2(x)]^k \cos^n(x) \sin(x) \ dx$ |
| Integral of even power of sine and cosine use trig identities | $\sin^2(x) = \frac{1}{2}(1-\cos(2x)) \newline \cos^2(x) = \frac{1}{2}(1+\cos(2x)) \newline \sin(x)\cos(x) = \frac{1}{2}\sin(2x)$ |
| Integral of even power of secant $(u = \tan x)$ |
$\int \tan^m(x)\sec^{2k}(x) \ dx \newline = \int \tan^m(x)[\sec^2(x)]^{k-1}\sec^2(x) \ dx \newline = \int \tan^m(x)[1+\tan^2(x)]^{k-1}\sec^2(x) \ dx$ |
| Integral of odd power of tangent $(u = \sec x)$ |
$\int tan^{2k+1}(x)\sec^n(x) \ dx \newline = \int[\tan^2(x)]^k\sec^{n-1}(x)\sec(x)\tan(x) \ dx \newline = \int [\sec^2(x)-1]^k\sec^{n-1}(x)\sec(x)\tan(x) \ dx$ |
| Trig identity for solving $\int \sin(mx)\cos(nx) \ dx$ |
$\sin A \cos B = \frac{1}{2}[\sin(A-B) + \sin(A+B)]$ |
| Trig identity for solving $\int \sin(mx)\sin(nx) \ dx$ |
$\sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)]$ |
| Trig identity for solving $\int \cos(mx)\cos(nx) \ dx$ |
$\cos A \cos B = \frac{1}{2}[\cos(A-B) + \cos(A+B)]$ |
Trigonometric substitution
| Expression | Substitution | Trigonometric Identity |
|---|---|---|
| $\sqrt{a^2 - x^2}$ | $x = a\sin\theta$ | $1 - \sin^2\theta = \cos^2\theta$ |
| $\sqrt{a^2 + x^2}$ | $x = a\tan\theta$ | $1+\tan^2\theta = \sec^2\theta$ |
| $\sqrt{x^2 - a^2}$ | $x = a\sec\theta$ | $\sec^2\theta-1 = \tan^2\theta$ |
Improper integrals
| Description | Equations |
|---|---|
| Improper integrals with single one-side infinite intervals | $\int_a^\infin f(x) \ dx = \lim\limits_{t\to\infin}\int_a^t f(x) \ dx \newline \int_{-\infin}^b f(x) \ dx = \lim\limits_{t\to-\infin}\int_t^b f(x) \ dx$ |
| Improper integrals with single two-side infinite intervals | $\int_{-\infin}^\infin f(x) \ dx \newline = \int_{-\infin}^a f(x) \ dx + \int_a^\infin f(x) \ dx$ |
| Convergence and divergence of improper integrals of power functions | $\displaystyle\int_1^\infin \dfrac{1}{x^p} \ dx$ convergent if $p>1$ divergent if $p \le 1$ |
| Improper integrals with discontinuous integrand on one side | $\int_a^b f(x) \ dx = \lim\limits_{t\to b^-}\int_a^t f(x) \ dx \newline \int_a^b f(x) \ dx = \lim\limits_{t\to a^+}\int_t^b f(x) \ dx$ |
| Improper integrals with discontinuous integrand in the middle $c$ | $\int_a^b f(x) \ dx = \int_a^c f(x) \ dx + \int_c^b f(x) \ dx$ |
| Comparison theorem $(f(x) \ge g(x) \ge 0, x \ge a)$ |
(a) If $\int_a^\infin f(x) \ dx$ is convergent, then $\int_a^\infin g(x) \ dx$ is convergent. (b) If $\int_a^\infin g(x) \ dx$ is divergent, then $\int_a^\infin f(x) \ dx$ is divergent. |
Applications of Integration
| Description | Equations |
|---|---|
| Areas between curves | $A = \int_a^b [f(x) - g(x)] \ dx$ |
| Volume by method of disks and washers | $V = \int_a^b A(x) \ dx$ |
| Volume by method of cylindrical shells (rotating about y-axis) |
$V = \int_a^b 2\pi x f(x) \ dx$ |
| Average value of a function | $\bar{f} = \frac{1}{b-a}\int_a^b f(x) \ dx$ |
| The mean value theorem of integrals | If $f$ is continuous on $[a,b]$, then there exists $c\in[a,b]$ such that $f(c) = \bar{f} = \frac{1}{b-a}\int_a^b f(x) \ dx$, $\int_a^b f(x) \ dx = f(c)(b-a)$ |
| Arc length formula | $L = \int_a^b \sqrt{1+[f'(x)]^2} \ dx$ |
| Arc length function | $s(x) = \int_a^x \sqrt{1+[f'(t)]^2} \ dt$ |
| Surface area of surface of resolution about x-axis | $S = \int_a^b 2\pi f(x)\sqrt{1 + [f'(x)]^2} \ dx$ |